Can I bake a christmas cake in a cardboard box?

While there is some possibility it could work, I wouldn't even try it. There are things that can go wrong, and many proven solutions to the problem. Replacing a bad hack (newspaper) with a worse hack (cardboard box) is not something I would be keen to try.

A list of preferred solution includes:

• use cake strips
• bake the cake in a different pan (bundt or thick-walled ceramic pan)
• go for a flatter and wider cake (bake it in a quiche pan or a large tawa)
• make it a layered cake and bake the layers separately
• reduce the temperature and bake for longer
• partially bake in a waterbath, then continue without it

If you are absolutely sure that your cake needs a full enclosure (which is quite strange for me - it is actually one of the reasons why I would avoid the cardboard box) I would prefer putting the cake in a roasting dish (earthenware/ceramic or metal) rather than in a cardboard box.

Although it's now clear that the question doesn't involve baking directly in the cardboard, but in using it as insulation, this answers the other interpretation of the title.

A cake cooked directly in cardboard should bake, as in cook through. It might take a little longer, which would make the top likely to go a bit too dark, and there are a few things you could do to make it better.

A thick cake mix (Christmas cake) would work better than a runny batter (anything light with oil), as it will soak into the cardboard less, though you should take steps to avoid that as well - I would line the box carefully with foil first. That will stop the cake tasting like cardboard, which I reckon is the biggest worry. The box getting wet would also weaken it, leading to a chance of dropping the cake, and by absorbing moisture it could make the outer layer rather dry.

Most corrugated boxes would leak down the middle underneath - another reason to line it, and the box may be held together with glue that could melt and let the cake escape, or smell horrible when scorched.

Personally I'd just use the most compatible proper container I already had, but have in the past made one-off baking forms from cardboard wrapped in foil (for shorter bakes than the typical Christmas cake).

I did a test where I split the cake batter in half, and put it into two identical tins. One was baked in the box and the other with no box and a foil hat on top. It was this recipe: https://helgavan.com/easy-christmas-cake-recipe/

And both were baked at 140°C for 3 hours.

Firstly, the box didn't fare too badly. I removed as much of the paper tape as possible. The remaining bits seemed to be completely unchanged. According to my research cardboard only spontaneously ignites > 400C, so it catching fire in the oven isn't a concern. As far as I can tell the water-based flexographic ink isn't toxic and won't break down after being exposed to heat for such a short period of time. But nonetheless, I probably won't do this again because the box isn't technically considered food-safe.

The box cake came out well. It was slightly darker around the edges but not burnt. It came out pretty much as expected. I've made this recipe yearly for several years and this was pretty much par for the course.

The no-box cake was significantly darker, not exactly burnt but closer than I would like. I did keep an eye on it but the cake batter is dark already so it is difficult to judge. The tin was fairly large and wide, so the cake wasn't that thick (5cm or so, i'd guess). I also did check on it at the 2h and 2.5h mark to check it wasn't done early, and it did need to go to 3h. I think next time I'll probably go the route of using a glass dish that has a larger thermal mass, and covering it with something more substantial than foil.